jmc

A $y$-intercept is a point on the graph that lies on the $y$-axis, so $x=0$. Hence, the number $y$-intercepts corresponds to the number of real solutions of the quadratic equation $y^2-4y-1=0$. The discriminant of this quadratic equation is $(-4{)}^2+4\cdot 1\cdot (-1)=20$, which is positive, so the quadratic has two distinct real roots. Therefore, the number of $y$-intercepts is $\boxed{2}$.